ELEMENTARY LECTURES ON ELECTRIC DISCHARGES, WAVES AND IlIPULSES, AND OTHER '~ritANSIENTS BY CHARLES PROTEUS STEINMETZ, A.M., PH.D. Pa:st P1·eside1it, American institute of Electrical Engineer3 SECOND ECDITION R&vt&Eu AND E:in.AROED McGltAW-HILL BOOK COMPANY, INc. 239 WEST 39TH STREET, NEW YORK 6 BOUVERIE STREET, LONDON, E. C. 1914 PREFACE TO THE SECOND EDITION. SINCE the issue of the first edition, in 1911, our knowledge of transients has greatly increased, and many of the phenomena, especially those of double energy transients and compound cir- cuits, have been observed and studied on transmission systems to a considerable extent, and have corroborated the oscillographic records given in the previous edition. • Considerable work has been done on momentary short circuits of alternators, and the variable component of the self-inductive reactance recognized as a transient reactance resulting from· the mutual induction of the armature with the field circuit. Especially in the field of sustained or continual, and of cumu- lative oscillations, a large amount of information has been gathered. The practical importance of these continual and cumulative oscil- lations has been strongly impressed upon operating and designing engineers in recen~ years, usually in the most disagreeable manner by the destruction of high power, high voltage transformers.- A chapter on these phenomena has therefore been added in the second edition. CHARLES P. STEINMETZ, A.M., PH.D. February, 1914. V PREFACE TO THE FIRST EDITION. IN the following I am trying to give a short outline of those phenomena which have become the most important to the electrical engineer, as on their understanding and control depends the further successful advance of electrical engineering. The art has now so far advanced that the phenomena of the steady flow of power are well understood. Generators, motors, transforming devices, transmission and distribution conductors can, with relatively little difficulty, be calculated, and the phenomena occurring in them under normal conditions of operation predetermined and controlled. Usually, however, the limitations of apparatus and lines are found not in the normal condition of operation, the steady flow of power, but in the phenomena occurring under abnormal though by no means unfrequent conditions, in the more or less transient abnormal voltages, currents, frequencies, etc.; and the study of the laws of these transient phenomena, the electric discharges, waves, and impulses, thus becomes of paramount importance. In a former work," Theory and Calculation of Transient Electric Phenomena and Oscillations," I hav.e given a systematic study of these phenomena, as far as our present knowledge permits, which by necessity involves to a considerable extent the use of mathematics. As many engineers may not" have the time or inclination to a mathematical study, I have endeavored to give in the following a descriptive exposition of the physical nature. and meaning, the origin and effects, of these phenomena, with the use· of very little and only the simplest form of mathematics, so as to afford a general knowledge of these phenomena to those engineers who have not the time to devote to a more extensive study, and also to serve as an introduction .to the study of " Transient Phenor{iena." I have, therefore, in the following developed these phenomena from the physical conception of energy, its storage and readjustment, and extensively used as illustrations oscillograms of such electric discharges, waves, and impulses, taken on industrial electric circuits of all kinds, as to give the reader a familiarity vii Vlll PREFACE 7'0 THE FIRST EDITION. with transient phenomena by the inspection of their record on the photographic film of the oscillograph. I would therefore recommend the reading of the following pages as an introduction to the study of "Transient Phenomena," as the knowledge gained thereby of the physical nature materially assists in the understanding of their mathematical representation, which latter obviously is necessary for their numerical calculation an'to the voltage given by· the generator. When the 1 2 ELECTRIC DISCHARGES, WAVES AND IMPULSES. condenser C is charged, the current in the line A and the condenser C is zero again. That is, the permanent condition before closing the switch S, and also some time after the closing of the switch, is zero current in the line. Immediately after the closing of the switch, however, current flows for a more or less short time. ·with the condition of the circuit unchanged: the same generator voltage, the switch S closed on the same circuit, the current nevertheless changes, increasing from zero, at the moment of closing the switch S, to a maximum, and then decreasing again to zero, ·while the condenser charges from zero voltage to the generator voltage. \Ve then here meet a transient phenomenon, in the charge of the condenser from a source of continuous voltage. A G A Fig. 2. Commonly, transient and permanent phenomena are superimposed upon each other. For instance, if in the circuit Fig. 1 we close the switch S connecting a fan motor F, at the moment of closing the switch S the current in the fan-motor circuit is zero. It rapidly rises to a maximum, the motor starts, its speed increases. while the current decreases, until finally speed arid current become constant; that is, the permanent condition is reached. The transient, therefore, appears as intermediate between two permanent conditions: in the above instance, the fan motor disconnected, and the fan motor running at full speed. The question then arises, why the effect of a change in the conditions of an electric circuit does not appear instantaneously, but only after a transition period, requiring a finite, though frequently very short, time. 2. Consider the simplest case: an electric power transmission (Fig. 3). In the generator G electric power is produced from ma. chanical power, and supplied to the line A. In the line A some of this power is dissipated, the rest transmitted into the. load L,, where the powe~ is used. The consideration of the electric power 4 ELECTRIC DISCHARGES, lVAVES AND IMPULSES. is the phenomenon by which the circuit readjusts itself to the change of stored energy. It may thus be said that the permanent phen9mena are the phenomena of electric power, the transients the phenomena of electric energy. 3. It is obvious, then, that transients are not specifically electrical phenomena, but occur with all forms of energy, under all conditions where energy storage takes place. Thus, when we start the motors propelling an electric car, a transient period, of acceleration, appears between the previous permanent condition of standst.ill and the final permanent condition of constant-speed running; when we shut off the motors, the permanent condition of standstill is not reached instantly, but a transient condition of deceleration intervenes. When we open the water gates leading to an empty canal, a transient condition-of flow and water level intervenes wbile the canal is filling, until the permanent condition is reached. Thus in the case of the fan motor in instance Fig. 1, a transient period of speed and mechanical energy appeared while the motor was speeding up and gathering the mechanical energy of its momentum. When turning on an incandescent lamp, the filament passes a transient of gradually rising temperature. Just as electrical transients may, under certain conditions, rise to destructive values; so transients of other forms of energy may become destructive, or may require serious consideration, as, for instance, is the case in governing high-head water powers. The column of water in the supply pipe represents a considerable amount of stored mechanical energy, when flowing at velocity, under load. If, then, full load is suddenly thrown off, it is not possible to suddenly stop the flow of water, since a rapid stopping .. would lead to a pressure transient of destructive value, that. is, burst the pipe. Hence the use of surge tanks, relief valves, or deflecting nozzle governors. Inversely, if a heavy load comes on suddenly, opening the nozzle wide does not immediately take care of the load, but momentarily drops the water pressure at the nozzle, while gradually the water column acquires velocity, that is, stores energy. The fundamental condition of the appearance of a transient thus is such a disposition of the stored energy in the system as differs from that required by the existing conditions of the system; and any change of the condition of a system, whicli requires a NATURE AND ORIGIN OF TRANSIENTS. 5 change of the stored energy, of whatever form this energy may be, leads to a transient. Electrical transients have been studied more than transients of other forms of energy because: (a) Electrical transients generally are simpler in nature, and therefore yield more easily to a theoretical and experimental investigation. (b) The theoretical side of electrical engineering is further advanced than the theoretical side of most other sciences, and especially: (c) The destructive or harmful effects of transients in electrical systems are far more common and more serious than with other forms of energy, and the engineers have therefore been driven by necessity to their careful and extensive study. 4. The simplest form of transient occurs where the effect is directly proportional to the cause. This is generally the case in electric circuits, since voltage, current, magnetic flux, etc., are proportional to each other, and the electrical transients therefore are usually of the simplest nature. In those cases, however, where this direct proportionality does not exist, as for instance .in inductive circuits containing iron, or in electrostatic fields exceed- ing the corona voltage, the transients also are far more complex, and very little work has been done, and very little is known, on these more complex electrical transients. Assume that in an electric circuit we have ~ transient cur- rent, as represented by curve i in Fig. 4; that is, some change of circuit condition requires a readjustment of the stored energy, which occurs by the flow of transient current i. This current starts at the value i1, and gradually dies down to zero. Assume now that the law of proportionality between cause and effect applies; that is, if the transient current started with a different value, ¼, it would traverse a curve i', which is the same as curve i, except that all values are changed proportionally, by the ratio -¼:- ; that I.S, t., = •t• X -¼:-• t1 t1 Starting with current i1, the transient follows the curve i; starting with i2, the transient follows the proportional curve i'. At some time, t, however, the current i has dropped to the value ¼, with which the curve i' started. At this moment t, the <;onditions in the first case, of current i, are the same as the conditions in 6 ELECTRIC DISCHARGES, WAVES AND IMPULSES. the. second case, of current i', at the moment t1; that is, from ( onward, curve i is the same as curve i' from time ti onward. Since 0 t1 t t 1 Fig. 4. - Curve of Simple Transient: Decay of Current. i' is proportional to i, from a.ny point t onward curve i is proportional to the same curve i from ti onward. At time t1, it is di2 = di1 X ~- dt1 dt1 i1 But sm• ce ddit-ii and 1•-2 at t1 are the same as ddti and i• at t·1me t, 1•t follows: or, d-i dt =dd-~t'1.1-ii1, di . dt = - ci, where c = - 1 ii ddi1t = constant, and t he minus sign is chosen, as ddti . 1s negat·1ve. As in Fig. 4: tan, if the flux1 interlinks the circuit n fold. I THE ELECTRIC FIELD. 13 the power p, which supplies the stored energy w of the dielectric field'¥. This current i' is called the:capacity curren< or, wrongly, charging current or condenser current. Since no power is required to maintain the field, but power is required to produce it, the capacity current must be proportional to the rate of increase of the dielectric field: ., i =dd-'Yt, (8) or by (6), i' = Cde. dt (9) If e and therefore 'I' decrease, :: and therefore i' are negative; that is, p becomes negative, and power is returned into the circuit. The energy supplied by the power p is w = JP df, (10) or by (7) and (9), w = J Cede; hence w =C2e-2 (11) is the energy of the dielectric field 'I'= Ce of the circuit. As seen, the capacity current is the exact analogy, with regard to the dielectric field, of the inductance voltage with regard to the magnetic field; the representations in the electric circuit, of the energy storage in the field. The dielectric field· of the circuit thus is treated and represented in the same manner, and ,vith the same simplicity and perspicuity, as the magnetic field, by using the same conception of lines of force. - • Unfortunately, to a large extent in dealing with the dielectric fields the prehistoric conception of the electrostatic charge on the conductor still exists, and by its use destroys the analogy between the two components of the electric field, the magnetic and the 14 ELECTRIC DISCHARGES, WAVES AND IMPULSES. dielectric, and makes the consideration of dielectric fields un- necessarily complicated. There obviously is no more sense in thinking of the capacity current as current which charges the conductor with a quantity of electricity, than there is of speaking of the inductance voltage as charging the conductor with a quantity of magnetism. But while the latter conception, together with the notion of a quantity of magnetism, etc., has vanished sinqe Faraday's representation of the magnetic field by the lines of magnetic force, the termi- nology of electrostatics of many textbooks still speaks of electric charges on the conductor, and the energy stored by them, without considering that the dielectric energy is not on the surface of..:..the conductor, but in the space outside of the conductor, just as the magnetic energy. 10. All the lines of magnetic force are closed upon themselves, all the lines of dielectric force terminate at conduct<;>rs, as seen in Fig. 8, and the magnetic field and the dielectric field thus can be considered as a magnetic circuit and a dielectric circuit. To produce a magnetic flux , a magnetomotive force Fis required. Since the magnetic field is due to the current, and is proportional to the current, or, in a coiled circuit, to the current times the num- ber of turns, magnetomotive force is expressed in current turns or ampere turns. F= ni. (12) If Fis the m.m.f., l the length of the magnetic circuit, energized byF, f = FT (13) is called the magnetizing force, or magnetic gradient, and is ex- pressed in ampere turns per cm. (or industrially sometimes in ampere turns per inch). In empty space, and therefore also, with very close approxi- mation, in all nonmagnetic material, f ampere turns per cm. length of magnetic circuit produce JC= 4 r f 10-t lines of magnetic force per square cm. section of the magnetic circuit. (Here the factor 10-1 results from the ampere being 10--1 of the absolute or cgs. unit of current.) JC= 4rf 10--l * • (14) * The factor 4 1r is a survival of the original definition of the magnetic field intensity from the conception of the magnetic mass, since unit magnetic mass was defined us that quantity of magnetism which acts on an equal quantity at THE ELECTRIC FIELD. 15 is called the magnetic-field intensity. It is the magnetic density, that is, the number of lines of magnetic force per cm2, produced by the magnetizing force off ampere turns p~r cm. in empty space. The magnetic density, in lines of magnetic force per cm2, produced by the field intensity :JC in any material is (15) whereµ is a constant of the material, a "magnetic conductivity," and is called the penneability. µ = 1 or very nearly so for most materials, with the exception of very few, the so-called magnetic materials : iron, cobalt, nickel, and some alloys and oxides of these metals and of manganese and chromium. If then A is the section of the magnetic circuit, the total ma~netic flux is cp = A = Li 108 lines of magnetic force. Inductance voltage: e'= n dd, JC, 0 = 8 X 106, and with 4 n total turns the total number of magnetic interlinkages thus is 4 ncl>0 = 32 n X 106, hence the inductance L = 4 ncl>0. 10-s = -.32.-n henrys. to to The field excitation is hence hence and ni0 = 6000 ampere turns, n =6-0.0-0 io L _ - .32 X 6000 t.o2 h enrys, T = L r = 1920 500 = 3.84 sec. That is, the stored magnetic energy could maintain full field excitation for nearly 4 seconds. It is interesting to note that the duration of the field discharge does not depend on the voltage, current, or size of the machine, but merely on, first, the magnetic flux and m.m.f., - which determine the stored magnetic energy, - and, second, on the excitation power, which determines the rate of energy dissipation. 15. Assume .now that in the moment where the transient begins the resistance of the coil in Fig. 10 is increased, that is, the i:::::ii:::::iiri.i-:::.n r lt:ltl:l:tttll L Fig. 12. - Magnetic Single-energy Transient. coil is not short-circuited upon itself, but its circuit closed by a resistance r'. Such would, for instance, be the case in Fig. 12, when opening the switch S. LECTURE_ II. THE ELECTRIC FIELD. 7. Let, in Fig. 7, a generator· G transmit electric power over line A into a receiving circuit lv[. While power flows through A ~ the conductors A, power is con- sumed in these conductors by lVI conversion in,to beat, repre- A sented by i'lr. This, however, Fig. 7. is not all, but in the space surrounding the conductor cer- tain phenomena occur: magnetic and electrostatic forces appear. Fig. 8. - Electric .Field of Conductor. The conductor is surrounded by a magnetic field, or a magnetic flux, which is measured by the number of lines of magnetic force 'q:i. With a single conductor, the lines of magnetic force are concentric circles, as shown in Fig. 8. By the return conductor, the circles 10 LECTURE III. SINGLE-ENERGY TRANSIENTS IN CONTINUOUSCURRENT CIRCUITS. 13. The simplest electrical transients are those in circuits in which energy can be stored in one form only, as in this case the change of stored energy can consist only of an increase or decrease; but no surge or oscillation between several forms of energy can exist. Such circuits are most of the low- and medium-voltage circuits,-· 220 volts, GOO volts, an r L ( ) charge of the Leyden jar. Fig. 10.-Mngnetie Single-energy Let, as represented in Fig. 10, Transient.. a continuous voltage eo be im- pressed upon a wire coil of resista.nce r and inductance L (but negligible capacity). A current io = ero flows through the coil and a magnetic field 'Po 10-s = Lnio. interlinks with the coil. Assuming now that the voltage eo is suddenly withdrawn, without changing 19 20 ELECTRIC DISCHARGES, WAVES AND IMPULSES. the constants of the coil circuit, as for instance by shortcircuiting the terminals of the coil, as indicated at A. With no voltage impressed upon the coil, and thus no power supplied to it, current i and magnetic flux of the coil must finally be zero. However, since the magnetic flux represents stored energy, it cannot instantly vanish, but the magnetic flux must gradually decrease from its initial value o, by the dissipation of its stored energy in the resistance o( the coil circuit as i 2r. Plotting, therefore, the magnetic flux of the coil as function of the time, in Fig. llA, the flux i::o~stant anlenoted by . Since the magnetic flux is pr~portional to the current, the latter must follow a curve proportional to , as shown in Fig. 11B. The impressed voltage is shown in Fig. 11C as a dotted line; it is e0 up to t 0, and drops to Oat to. However, ~ince after t0 a current i flows, an e.m.f. must exist in the circuit, proportional to the current. e = ri. SINGLE-ENERGY TRANSIENTS. 21 This is the e.m.f. induced by the decrease of magnetic flux 0, and the current i thus also full value i 0• Hence, at the first moment of short circuit, the induced e.m.f. e must be equal to e0, that is, the magnetic flux

cf>oE - c (t - to), = i ioe-c, i, and e, the total area of the voltage curve must be the same, and equal to n = lines of magnetic force. l\fognetomotive force: F = ni nrnpere turns. Permeance: Dielectric flux (dielectric current): it = lines of dielectric force. • Electromotive force: e = volts. Electric current: i = electric cur- rent. Voltnge: e = volts. ill=_!_· .41rll Inductance: L= n;:10-s= n:10-s Permittance or capacity: Conductance: C = !e farads. g = :e mhos. henry. Reluctance: R= !· Magnetic energy: w = L 2 i2 = F,f> 2 1 0-s joules. l\fagnetic density: CB= ,Af> =µJCliuespcrcm 2• Magnetizing force: f = TFampere turns per (Ela.stance): c 1 = e ii. Dielectric energy: w = Cc 2 2 = 2c'1t.Joules. Dielectric density: D =~=,cf( 1ines per cmi. Dielectric gradient: G = le volts per cm. Resistance: r =~ohms. t Electric power:. ' p = ri2 = gc2 = ei • watts. Electric-current density: I= Ai = -yGam- perespercm2•. Electric gradient: G = le volts per cm. cm. Magnetic-field intensity: Dielectric-field sity: JC= .41rf. K=4-1Gr1v2 09• in ten- . Permeability: µ = (B. JC Permittivity or specific Conductivity: capo.city: ,c= KD . -y = GI mho-cm. Reluctivity: p = L. (B (Elastivity ?): ; l = K D. Resistivity: p 1 =;= 1Gohm-cm. Specific mngnetic energy: Specific dielectric energy: Specific power: ,,:, = = tl)o .4 ,rµf.:. = f<9., 10-s = 2 •2 = Wo /((JI. 4- 1d 09 GD -21 0 9 = cl[$ S1r 10-7 J. oulespercm 3• 2 1rrPKD joules per cm3• = = = Po pP. (JI. GI watts per cm3. SINGLE-ENERGY· TRANSIENTS. 31 sient, shown clotted in Fig. 15. Adding the transient current io to the permanent curre~t ·i2 gives the total current during the transition period; which is shown in drawn line in Fig. 15. As seen, the transient is clue to the difference between the instantaneous value of the current i1 which exists, and that of the current i2 which should exist at the moment of change, and Fig. 15. - Single-energy Transient of Alternating-current Circuit. thus is the larger, the greater the difference between the two currents, the previous and the after current. It thus disappears if the change occurs at the moment when the two currents 1·1 and i2 are equal, as shown in Fig. 15B, and is a maximum, if the change occurs at the moment when the two currents i 1 and iz have the greatest difference, as shown in Fig. 15C, that is, at a point one-quarter period or 90 degrees distant from the intersection of i1 and i2, SINGLE-ENERGY. TRANSIENTS. 33 half the value of is0, and are opposite in direction thereto. In any case, the three transients must be distributed on both sides of the zero line. This is obvious: if it', 't2', and is' are the instan- taneous values of the permanent three-phase currents, in Fig. 17, the initin.I values of their tra.nsients are: -it', - i2', -is'. A il Fig. 17. - Single-energy Starting Transient of Three-phase Circuit. Since the sum of the three three-phase currents at every moment is zero, the sum of the initial values of the three transient currents also is zero. Since the three transient curves i1°, i,i.°, i3° are proportional to each other (as exponential curves of the same dura- tion T = Lr ), and the sum of their initial values is .zero, it follows 22 ELECTRIC DISCHARGES, WAVES AND IMPULSES. shown dotted in Fig. 11G. The area of this new voltage curve would be eoT, and since it is the same as that of the curve e, as seen above, it follows that the area of the voltage curve e is 'Zet = e~T, } = rioT, (3) and, combining (2) and (3), io cancels, and we get the value of T: T = -Lr· (4) That is, the initial decrease of current, and· therefore of mag• netic flux and of induced •voltage, is such that if the decrease continued at the same rate, the current, flux, and voltage would become zero after the time T = !r:. • The total induced voltage, that is, voltage times time, and therefore also the total current and magnetic flux during the transient, are such that, when maintained at their initial value, they would last for the time T = !r:. • Since the curves of current and voltage theoretically never become zero, to get an estimate of the duration of the transient we may determine the time in which the transient decreases to half, or to one.:.tenth, etc., of its initial value. It is preferable, however, to estimate the duration of the transient by the time T, which it would last if maintained at its initial value. That is, the duration of a transient is considered as the time T = !r:. • This time T has frequently been called the " time constant " of the circuit. The higher the inductance L, the longer the transient lasts, obviously, since the stored energy which the transient dissipates is proportional to L. The higher the resistance r, the shorter is the duration of the transient, since in the higher resistance the stored energy is more rapidly dissipated. . Using the time constant T = !r:. as unit of length for the abscissa, and the initial value as unit of the ordinates, all exponential transients have the same shape, and can thereby be constructed SINGLE-ENERGY TRANSIENTS. 37,. apparatus, however, these momentary starting currents usually are far more limited than in transformers, by the higher stray field (self-inductive reactance), etc., of the apparatus, resulting from the air gap in the magnetic circuit. 19. As instance of the use of the single-energy transient in engineering calculations may be considered the investigation of the momentary short-circuit phenomena of synchronous alternators. In alternators, especially "high-speed high-power machines as turboalternators, the momentary short-circuit current may be many times greater than the final or permanent shortcircuit current, an = 4>oE- ct = 4>oE-T = 4>oE-1,, t rt i = ioE- ct = ioE- T = ioE- L, (6) t rt e = eoE-ct = eoE- T == eoE- L. The same equations may be derived directly by the integration of the differential equation: + . L ddti 0 ri:::::a' (7) where L :: is the inductance voltage, ri the resistance voltage: and their sum equals zero, as the coil is short-circuited. Equation (7) transposed gives di i = - Lr dt, hence L logi = - t + logC, _!:. t i = CE L' and, as for t = 0: i = io, it is: C = io; hence 14. Usually single-energy transients last an appreciable time, and thereby become of engineering importance, only in highly inductive circuits, as motor fields, magnets, etc. To get an idea on the duration of such magnetic transients, consider a motor field: A 4-polar motor haso:8 ml. (megalines) of magnetic flux per pole, produced by 6000 ampere turns m.m.f. per pole, and dissi- pates normally 500 watts in the field excitation.. • That is, if io = field-exciting current., n = number of field turns per pole, r = resistance, and L = inductance of the field-exciting circuit, it is io2r = 500, hence r--5-io020· SINGLE-ENERGY TRANSIENTS. 89 Thus it is: momentary short-circuit current open-circuit field flux * permanent short-circuit current = short-circuit field flux = armature reaction plus self-induction synchronous reactance xo self-induction = self-inductivereactance = X1 • 20. Let ¢ 1 = field flux of a three-phase alternator (or, in general, polyphase alternator) at open circuit, and this alternator be short- circuited at the time t = O. The field flux then gradually dies o, as indicated by the curve in Fig. 21A. If m = ratio armature reaction plus self-induction armature self-induction it is 1 = m¢o, and the initial value of the field flux consists of the permanent part cI>o, and the transient part ' = 1 -<1>0 = (m-1) tl>0• This is a rather slow trn.nsient, frequently of a duration of a second or more. The armature currents ii, i~, ia are proportional to the field flux 1 is higher than 0, or m times, and are represented in Fig. 21B. The resultant m.m.f. of the armature currents, or the armature reaction, is proportional to the currents, and thus follows the same field transient, as shown by F in Fig. 21 C. The field-exciting current is i0 at open circuit as well as in the permanent condition of short circuit. In the permanent condition of short circuit, the field current i 0 combines with the armature reaction F0, which is demagnetizing, to a resultant m.m.f., which produces the short-circuit flux ¢0. During the transition period the field flux is higher than tl>o, and the resultant m.m.f. must therefore be higher in the same proportion. Since it is the difference between the field current and the armature reaction F, and the latter is proportional to , the field current thus must also be * If the machine were open-circuited before the short circuit, otherwise the field flux existing before the short circuit. It herefrom follows that the momentary short-circuit current essentially depends on the field flux, and thereby the voltage of the machine, before the short circuit, but is practically independent of the load on the machine before the short circuit and the field excitation corresponding to this load. 26 ELECTRIC DISCHARGES, WAVES AND IMPULSES. The transients of magnetic flux, current, and voltage are shown as A, B, and C in Fig. 13. The magnetic flux and therewith the current decrease from the initial values o and io at the moment t0 of opening the switch S, on curves which must be steeper than those in Fig. 11, since the current passes through a greater resistance, r + r', and thereby dissipates the stored magnetic energy at a greater rate. o I I I I A I I I 1o I B I I I I I e'_I __ _ _ _ _ _ I 0 I I I -~-- e I I C ' tI o t Fig. 13. - Characteristics of Magnetic Single-energy Transient. The impressed voltage e0 is withdrawn at the moment t0, and a voltage thus induced from this moment onward, of such value as to produce the current i through the resistance r + r'. In the first moment, t0, the current is still io, and the induced voltage thus must be eo' = io (r + r'), while the impressed voltage, before to, was eo = ior; hence the induced voltage eo' is greater than the impressed volt- age e0, in the same ratio as the resistance of the discharge circuit r + r' is greater than the resistance of the coil r thri::1ugh which the impressed voltage sends the current eo' e0 =r-+r-r'· SINGLE-ENERGY TRANSIENTS. 43 in Fig. 21C by F. During the initial part of the short circuit, however, while the armature transient is appreciable and the armature currents thus unsymmetrical, as seen in Fig. 22B, their ·resultant polyphase m.m.f. also shows a transient, the transient of the rotating magnetic field discussed in paragraph 18. That is, it approaches the curve F of Fig. 21C by a series of oscillations, as indicated in Fig. 21E. ' Since the resultant m.m.f. of the machine, which produces the flux, is the difference of the field excitation, Fig. 21D and the armature reaction, then if the armature reaction shows an initial os- cillation, in Fig. 21E, the field-exciting current must give the same oscillation, since its m.m.f. minus the armature reaction gives the resultant field excitation corresponding to flux cf?. The starting transient of the polyphase armature reaction thus appears in the field current, as shown in Fig. 22C, as an oscillation of full machine frequency. As the mutual induction between armature and field circuit is not perfect, the transient pulsation of armature reaction appears with reduced amplitude in the field current, and this reduction is the greater, the poorer the mutual inductance, that is, the more distant the field winding is from the armature wind- ing. In Fig. 22C a damping of 20 per cent is assumed, _which corresponds to fairly good mutual inductance between field and armature, as met in turboalternators. If the field-exciting circuit contains inductance outside of the alternator field, as is always the case to a slight extent, the pul- sations of the field current, Fig. 22C, are slightly reduced and delayed in phase; and with considerable inductance intentionally inserted into the field circuit, the e:ffect of this inductance would require consideration. From the constants of the alternator, the momentary short- circuit characteristics can now be constructed. Assuming that the duration of the field transient is .To = ( m Lo - 1) To = 1 sec., the duration of the armature transient is T = -Lr = .I sec. .. And assuming that the armature reaction is 5 times the armature 28 ELECTRIC DISCHARGES, WAVES AND IMPULSES. The same reasoning also applies to the transient resulting from several forms of energy storage (provided that the law of proportionality ()f i, e, , and voltage e on the coil are zero. In final condition, after the transient has passed, the values io, 4>o, eo are reached. We may then, as discussed above, separate the transient from the permanent term, and consider that at the time to the coil has a permanent current io, permanent flux cl>o, permanent voltage e0, and in adcli- SINGLE-ENERGY 'l'RANSIENTS. 45 on the point of the wave at which the phenomenon begins, but not so in their resultant effect. • 21. The conditions with a single-phase short circuit are differ- ent, since the single-phase armature reaction is pulsating, vary- ing between zero and double its average value, with double the machine frequency. The slow field transient and its effects are the same as shown in Fig. 21, A to D. However, the pulsating armature reaction produces a corre- sponding pulsation in the field circuit. This pulsation is of double .1 .2 .4 Seconds A B C lo Fig. 23. -Symmetricn.l Moment0,ry Single-phase Short Circuit of Alternator. frequency, and is not transient, but equally exists in the final shortcircuit current. Furthermore, the annature transient is not constant in its reaction on the field, but varies with the point of the wave at which the short circuit starts. Assume that the short circuit starts at that point of the wave where the permanent (or rather slowly transient) armature current should be zero: then no armature transient exists, and the armature current is symmetrical from the beginning, and shows the slow transient of the field, as shown in Fig. 23, where A . LECTURE IV. SINGLE-ENERGY TRANSIENTS IN ALTERNATINGCURRENT CIRCUITS. I7. Whenever the conditions of an electric circuit are change~ in such a manner as to require a change of stored energy, a transition period appears, during which the stored energy adjusts itself from the condition existing before the change to the condition after the change. The currents in the circuit during the transition period can be considered as consisting of the superposition of the permanent current, corresponding to the conditions after the change, and a transient current, which connects the current value before the change with that brought about by the change. That is, if i1 = current existing in the circuit immediately before, and thus at the moment of the change of circuit condition, and i2 = current which should exist at the moment of change in accordance with the circuit condition after the change, then the actual current i1 can be considered as consisting of a part or component i2, and a component i1 - i2 = io, The former, i2, is permanent, as result- ing from the established circuit condition. The current component io, however, is not produced by any power supply, but is a remnant of the previous circuit condition, that is, a transient, and therefore gradually decreases in the manner as discussed in para- graph 13, that is, with a duration T = .rf. • = G,L. The permanent current i2 may be continuous, or alternating, or may be a changing current, as a transient of long duration, etc. The same reasoning applies to the voltage, magnetic flux, etc. Thus, let, in an alternating-current circuit traversed by current i1, in Fig. 15A, the conditions be changed, at the moment t = 0, so as to produce the current i2• The instantaneous value of the current i1 at the moment t = 0 can be considered as consistin.g of the instantaneous value of the permanent current ¼, shown dotted, and the transient io = i1 - i2, The latter gradually dies down, with the duration T = Lr , on the usual exponential tran- 30 49 frequenry, and nf'< tho r0sult; an incn•ase of volt.age a111l u disfort.ion of t.lie quadrature phase occur:-,, ns shown in the uscillogram Fig. 2G. Various morneutary short-circuit phenomena are illustrated by the oseillogrnms Figs. 26 to 28. Figs. 2011 und 2GB show the momentary three-phase short cir~uit of n 4-polnr 2G-cycle 1500-kw. steam t,urhine alternator. The FiJ,!:. 20. I. - cD!l:1\l!l. - Rymmetrical. Fig. 2fiR. - rn030i. - Asymmetrical. l\Iomeutnry Thn•<'-pha"'" 1--:hort Cimiit, or 1300-Kw. 2300-Volt -11uce-phnse Alt erna1.nr (A Tn--1- HiOO-- I SOil). 0!4dllograrni:; of Arnmture Current am l Fidu Current. lower cun·' = ~b • i If r = resistance, the duration of the component of the transient resulting from the air flux would be T2 = L-rt = nc 10-s - -r - , (5) and the duration of the transient which would result from the initial inductance of the iron flux would be Ti= L1 = na10-s_ r r (6) The differential equation of the transient is: induced voltage plus resistance drop equal zero; that is, n c: 10-s + ri = 0. Substituting (3) and differentiating gives (·n1·-a+-1-·0b--i·)s-2 -ddti + nc 10-s -ddti + ri. = 0' an. Thus, as it is i = i0 at cJ>0, during the transition period it is i = i0• Hence, the field-exciting current traverses 0 the same transient, from an initial value io' to the normal value i 0, as the field flux and the armature currents. B - ··---~ ---...... --- - ----- - ---- --- -- - C 0 Fig. 21. - Construction of Momentary Short Circuit Chamcteristic of Polyphase Alternator. Thus, at the moment of short circuit a sudden rise of field current must occur, to maintain the field flux at the initial value 4>1 against the demagnetizing armature reaction. In other words, the field flux decreases at such a rate as to induce in the field circuit the e.m.f. required to raise the field current.in the proportion m, from i0 to io', and maintain it at the values corresponding to the transient i, Fig. 21D. As seen, the transients ; ii, i2, i3 ; Fi i are proportional to each other, and are a field transient. If the field, excited by current io .· DOUBLE-ENERGY TRANSIENTS. 61 energy is dissipated before this. This latter case occurs when the dissipation of energy is very rapid, the resistance (or conductance) high, and therefore gives transients, which rarely are of industrial importance, as they are of short duration and of low power. It therefore is sufficient to cousider the oscillating 1 A i1 B C 1 lo Fig. 22. - 11omentary Short Circuit Characteristic of Three-phase Alternator. currents for the case where the circuit is closed at the moment when i1 should be maximum; i1 then shows the maximum transient, and i2 and ia transients in opposite direction, of half amplitude. These armature transients rapidly disappear, and the three currents become symmetrical, and gradually decrease with the field transient to the final value indicated in the figure. The resultant m.m.f. of three three-phase currents, or the armature reaction, is constant if the currents are constant, and as the currents decrease with the field transient, the resultant armature reaction decreases in the same proportion as the field, as is shown DOUBLE-ENERGY TRANSIEN'J.'S. 63 oscillating voltages, that is, acts as a short circuit for the trans- former oscillation, and therefore protects the latter. Inversely, if the large oscillating current of a cabl~ enters a reactive device, as a current transformer, it produces enormous voltages therein. Thus, cable oscillations arc more liable to be destructive to the reactive apparatus, transformers, etc., connected with the cable, than to the cable itself. A transmission line is intermediate in the values of z0 and Yo between the cable and the reactive apparatus, thus acting like a reactive apparatus to the former, like a cable toward the latter. Thus, the transformer is protected by the transmission line in oscillations originating in the transformer, but endangered by the transmission line in oscillations originating in the transmission line. v§ The simple consideration of the re;~tive values of Zo = in the different parts of an electric system thus gives considerable information on the relative danger and protective action of the parts on each other, and shows the reason why some elements, as current transformers, are far more liable to destruction than others; but also shows that disruptive effects of transient voltages, observed in one apparatus, may not and very frequently do not originate in the damaged apparatus, but originate in another part of the system, in which they were relatively harmless, and become dangerous only when entering the former apparatus. 26. If there is a periodic transfer between magnetic and dielectric energy, the transient current i ancl the transient voltage e successively increase, decrease, and become zero. The current thus may be represented by i = io cos(¢, - -y), (12) where i0 is the maximum value of current, discussed above, and = ¢, 2 1r ft, (13) where f = the frequency of this transfer (which is still undeter- mined), and -y the phase angle at the starting moment of the +,ransient; that is, i1 = io cos -y = initial transient current. (14) As the current -i is a maximum at the moment when the magnetic ,.,nergy is a maximum and the dielectric energy zero, the voltage e '44 ELECTRIC DISCIIARGES, WAVES AND IlvfP.ULSES. self-induction, that is, the synchronous reactance is 6 times the self- inductive reactance, Xo = m = 6. The frequency is 25 cycles. Xi If cI>1 is the initial or open-circuit flux of the machine, the short- ! circuit flux is cI>o = cI>i = cI>1, an is a tran1n u sient of duration 1 sec., connecting cI>1 and o, Fig. 22A, represented by the expression t c.I> = cl>o + (cI>1 - o)E- To. The permanent armature currents ii, i2, i3 then are currents starting with the values 1n eo , and decreasing to the final short- Xo circuit current eo, on the field transient of duration To, To these Xo currents are added the armature transients, of duration T, which start with initial values equal but opposite in sign to the initial values of the permanent (or rather slowly transient) armature currents, as , starting with io' = mio, and tapering to the final value io, Upon this is superimposed the initial full-frequency pulsation of the armature reaction. The transient of the rotating field, of duration. T = .1 sec., is constructed as in paragraph 18, and for its instan- taneous values the percentage deviation of the resultant field from its permanent value is calculated. Assuming 20- per cent damping in the reaction on the field excitation, the instantaneous values of. the slow field transient (that is, of the current (i - i0), f;ince io is the permanent component) then are increased or decreased by 80 per cent of the percentage variation of the transient field of armature reaction from uniformity, anc.l thereby the field curve, Fig. 22C, is derived. Here the correction for the external field inductance is to be applied, if considerable. Since the transient of the armature reaction does not depend on the point of the wave where the short circuit occurs, it follows that the phenomena at the short circuit of a polyphase alternator are always the same, that is, independent of the point of the wave at which the short circuit occurs, with the exception of the initial wave ~hape of the armature currents, which individually depend . DOUBLE-ENERGY TRANSIENTS. 65 decreases, an (the same as in Fig. 22A) and B the armature current, decreasing from an initial value, which is m. times the final value, on the field transient. Assume then that the mutual induction between field and armature is such that 60 per cent of the pulsation of armature reaction appears in the field current. Forty per cent damping for the double-frequency reaction would about correspond to the 20 per cent damping assumed for the transient full-frequency pulsation of the polyphase machine. The transient field current thus pulsates by 60 per cent around the slow field transient, as shown by Fig. 23C; passing a maximum for every maximum of armature .1 .2 .4 Seconds A B C Fig. 24. -Asymmetrical Momentary Single-phnse Short Circuit of Alternator. current, and thus maximum of armature reaction, and a minimum for every zero value of armature current, and thus armature reaction. Such single-phase short-circuit tmnsients have occasionally been recorded by the oscillograph, as shown in Fig. 27. Usually, however, the circuit is closed at a point of the wave where the permanent annature current would not be zero, and an armature transient appears, with an initial value equal, but opposite to, the initial value of the permanent armature current. This is shown in Fig. 24 for the case of closing the circuit at the moment where the. DOUBLE-ENERGY TRANSIENTS.• 67 transient. In the lat'ver case, the duration of the transient would be To=L-r , and with only half the energy magnetic, the duration thus is twice as long, or (23) and hereby the factor t h = E-T1 multiplies with the values of current anii I'..!S. -- HyrnmPl.ri1·:d. i\lo11wnlary Single-phase Short Circuit of /\ It Prnat.or. ( lP.cillog;ra111 of An11al 111•p ( '.11rrc111'., Armature Voltage, :llld FiPld ( '111TP11I. (C'ircnil hrC'akPr opens.) Fi~. 2X. - c1>1ii;1>ii. -- ,\,-ym111Plri(·:t I. i\lomcutary 8ingle-phusc 8hort Circuit of ii000-1\.w. 11,0110-Vnlt ThrPe-phase ..\11.ernator (A'rB-6-5000-500). Uscillogrnm of Armature Current uml Ficlll Current. Fig. 28 shows the single-phase short circuit of a 6-polar 5000-kw. 11,000-Yolt steam turbine alternator, which occurred at a point of th<' wm·P. wlwrc the armat.11n~ current should be not far from its rnaxmm111. The t.nmsiPnt armat,urc currC'nt, therefore, st.arts un- DOUBLE-ENERGY 1'RANSIE1V'l'S. 71 Fig. 33A gives the periodic components of current and voltage: i' = 140 cos 0.2 t - SO sin 0.2 t, e' = 2000 cos 0.2 t + 3500 sin 0.2 t. Fig. 33B gives = The magnetic-energy transient, h i:- 1, The dielectric-energy transient, k = E- 2!, · And the resultant transient, hk = E-a:. And Fig. 33C gives the transient current, i = hki', and the transient voltage, e = hke'. LECTURE V. SINGLE-ENERGY TRANSIENT OF IRONCLAD CIRCUIT. 22. Usually in electric circuits, current, voltage, the magnetic field and the dielectric field are proportional to each other, and the transient thus is a simple exponential, if resulting from one form of stored energy, as discussed in the preceding lectures. This, however, is no longer the case if the magnetic field contains iron or other magnetic materials, or if the dielectric field reaches densities beyond the dielectric strength of the carrier of the fiel - 'Y) (11) 54 ELECTRIC DISCHARGES, WAVES .tlND IMPULSES. or, substituting gives equation (1). cB1 ' = a, 1 (J?, , = u' C 00 00 For JC = 0 in equation (1), JCO?, = ! ; for JC = oo, O?, = ! ; that is, a u m• equat·1011 (1), a-1 = n• u•tm• l permeab1'l1' ty, -u1 = saturati•on value of magnetic density. If the magnetic circuit contains an air gap, the reluctance of the iron part is given by equation (2), that of the air part is constant, and the total reluctance thus is p = fJ + a-JC, where (3 = a plus the reluctance of the air gap. Equation (1), therefore, remains applicable, except that the value of a is increased. In addition to the metallic flux given by eguation (1), a greater or smaller part of the flux always passes through the air or through space in general, antr. Fig. 36. - en10002. -Oscillogmm of Sturting Oscillation of 28 Mile:1 of 100,000-\·olt Transmission Line: Lowtension Switching. -1 -1 58 ELECTRIC DISCHARGES, WAVES AND IMPULSES. This gives T1 = 4, T2 = .4. .Assuming io = 10 amperes for t0 = 0, gives from (10) the equa- tioo: • j ~ T = 2.92 - 9.21 log10 l +i.6i + .921 log10 i + l :.6i • Herein, the logarithms have been reduced to the base 10 by division with log10E = .4343. For comparison is shown, in ugt.h the tfo;tancc of propagation in unit tinw, or :3 X 1010 cm. in uv1•rhea - 'Y) l, = e1 eo sin ( - 'Y) ~ ' (3) where = 2 1rft, (4) and 1 J= 21rvLC (5) is the frequency of oscillation. The dissipative or " transient " component is = hk E-u1, (6) 72 TRAVELING lVAVES. 98 the rate e-u,, corresponding to the dissipation of the stored energy by e-" 1, as indicated by A' in Fig. 42; while in the case (b) the power flow decreases faster, in case. (c) slower, than corresponds to the energy dissipation, an into the equation (11), the equations of the line oscillation: i = cc"t cos ( =t= w - -y) } = e ZoCE-ut sin (- e~:OE-2u.tE+2s(t->-). 95 (12) Both forms of the expressions of l:, e, and J)o of equations (11) and (12) are of use. The first form shows that the wave de- creases slower with the time t, but decreases with the distance X. The second form shows that the distance X enters the equation only in the form t - X and .. Fur a wave traveling in opposite direction, the sign of X and thus of w is reversed. (e) If the now of power increases along tho line, more power leaves every line eleuH'nt than enters it; that is, the line clement is drained of its stored energy by the passage of the wave, and thus the transient dies down with the time at a greater rate than corre- spomls to the power dissipation.by r and g. That is, not all the stored energy of the line elements supplies the power which is being dissipated in the line element, but a part of the energy leaves the line element in increasing the power which flows along the line. The rate of dissipation thus is increased, and instead of u, (u + s) enters the equation. That is, the e1q)onential time decrement is E-(u+ s)t (13) ' but inversely, along the line X the power flow increases, that is, the intensity of the wave increases, by the same factor E+s>-, or rather, the wave decreases along the line at a slower rate than corresponds to the power dissipation. The equations then become: i i=ioE-cu+s)tE+.•>-cos (-)cos (t E+s>- cos(-> cos (. e~o E-2s(t-;>,) 1 (15) iG r Ji)[,lff7TNIC /)/,'WI/ AR<:ES, 1r.,1 HS .1lN I) IM f>UL8ES. TRAVELJ;'VG WAVES. 97 increase with the time, which in general is not possible; as the traPsient must decrease with the time, by the power dissipation in rand g. Standing waves and traveling waves, in which the coefficient in the exponent of the time exponential is positive, that is, the wave increases with the time, may, however, occur in electric circuits in which the wave is supplied with energy from some outside source, as by a generating system flexibly connected (electrically) through an arc. Such waves then are "cumulative oscillations." They may either increase in intensity indefinitely, that is, up to destruction of the circuit insulation, or limit themselves by the power dissipation increasing with the increasing intensity of the oscillation, until it becomes equal to the power supply. Such oscillations, which frequently u.re most destructive ones, are met in electric systems as "arcing grounds," "grounded phase," etc. They are frequently called "undamped oscillations," and as such find a use in wireless telegraphy and telephony. Thus far, the only source of cumulative oscillation seems to be an energy supply over an arc, especially an unstable arc. In the self-limiting cumulative oscillation, the so-called damped oscillation, the transient becomes a permanent phenomenon. Our theoretical knowledge of the cumulative oscillations thus far is rather limited, however. • An oscillogram of a "grounded phase " on a 154-mile threephase line, at 82 kilovolts, is given in Figs. 44 and 45. Fig. 44 shows current and voltage at the moment of formation of the ground; Fig. 4.5 the same one minute later, when the ground was fully developed. An oscillogram of a cumulative oscillation in a 2500-kw. 100,000volt power transformer (GO-cycle system) is given in Fig. 46. It is caused by slvitching off 28 miles of line by high-tension switches, at 88 kilovolts. As seen, the oscillation rapidly increases in in- tensity, until it stops by the arc extinguishing, or by the destruction of the transformer. Of special interest is the limiting case, - s = u; in this case, u + s = 0, and the exponential function of time vanishes, and current and voltage become i = ioe:!:a>. cos(¢ =F w - 1'), ( e = Coe=">. cos (¢ =F w - -y), ~ (18) 18 ELECTRIC DISCHARGES, WA YES AND IMPULSES. Instead of L and C, thus enter into the equation of the double- energy oscillation of the line the values 2 L and 2 C.. • "If 7r 7r In the same manner, instead of the total resistance r and the total conductance g, the values -2r and -2g appear. " 7r 7r The values of zo, y0, u, ¢, and w are not changed hereby. The frequency f, however, changes from the value correspond- ing to the circuit of massed capacity, f = ~ , to the value 21r LC ! -- 4--V1-LC· Thus the frequency of oscillation of a transmission line is 1 1 f = 4 VW = 4 u' (20) where (21) If l1 is the length of the line, or of that piece of the line over which the oscillation extends, and we denote by Lo, Co, ro, (Jo (22) the inductance, capacity, resistance, and conductance per unit length of line, then u = ½(~: + i:); (23) that is, the rate of decrease of the transient is independent of the length of the line, and merely depends on the line constants per unit length. It then is (24) where (25) is a constant of the line construction, but independent of the length of the line. The frequency then is (26) ~ .~... ~ l:;1'j .t.-.-. ~ ~ ..... :~:. ~ ~ 0 Fig. 46.-cnlO0Ol.-Oscillogrum of Cumulative OsciUation in High-potential Coil of 2500-kw. Step-up Transformer· Caused by Disconnecting 28 Miles of 100,000-volt Transmission Line; High-temlion Switching. <:o (Cl 80 •ELECTRIC JJISCIIARGES, WAVES ,1ND /1v!PULSES. um•t of osc1·11at1•on 1•s 311 , or a1so a quarter-wave. The same 1•s th? case in Fig. 37C, etc. In the case 2, i = 0 at both ends of the line, the current and voltage distribution are as sketched in Fig. 38, A, B, C, etc. That is, in A, the section l1 is a half-wave, but the middle, C, of l1 is a node or point of zero power, and the oscillating unit again is a quarter-wave. In the same way, in Fig. 38B, the section l1 consists of 4 quarter-wave units, etc. rI' ---e---..__._ I Al I I I I I ... _ I'I , ....._ , II IA I I I I ' I ',I Bl I I I ' 1 / I _,..- I I I ' I ' ', l C,',... Bl I ',.._ I --, ,, I .._.., I I '-, I '--+ I Bl ! I ' ' iI I Fig. 37. Fig. 38. The same applies to case 1, and it thus follows that the wave length lo is four times the length of the oscillation l1. 30. Substituting lo = 4 l1 into (26) gives as the frequency of oscillation f =lo-1110· (30) However, if f = frequency, and v = ! , a velocity of propagation, the wave length lo is the distance traveled during one period: to_= J1 = per1•0d, (31) TRAVELING WAVES. 101 When traveling waves and stationary waves occur simultaneously, very often the traveling wave prec1::1fos the stationary wave. The phenomenon may start with a traveling wave !)r impulse, and this, by reflection at the ends of the circuit, and ·COflir.>ination of the reflected waves aml the main waves, gradually chai>gt;s.-to a stationary wave. In this case, the traveling wave has the sai~o frequency as the stationary wave resulting from it. In Fig. 47'is shown the reproduction of an oscillogram of the formation of a stationary oscillation iu a transmission line by the repeated re- '<......._:.: ____;;:..> :::::::,.., <::..:...:..:: Fig. 47. -col 1168. - Reproduction of :i.n Oscillogmm or Stationary Line Oscillation by Reflection of Impulse from Ends or Line. (The lowest curve gives a 6()..cycle current as time mea::;ure.) flection from the ends of the line of the single impulse caused by short circuiting the energized line at one en denotes the time angle and w the distance angle of the fundamental wave, that is, = 2 1r represents a complete cycle and w = 2 1r a complete wave length of the fundamental wave, the time and distance angles of the higher harmonics are ;3 , 3 w, 5 , fi w, 7 , 7 w, etc. A complex oscilln.tion, compr1smg waves of all possible frequencies, thus would have the form + a1 cos ( =i= w - 1'1) 0.3 cos 3 ( =F w - 'Ya) + a,, cos i'5 ( =F w - + -y5) . . . , (37) and the length Z1 of the line then is represented by the angle w = ; , arnl the oscillation called a qnarter-wave oscalation. If the t,vo ends of the line Z1 have the same electrical charac- t<'rh;;t.ics, that is, e = 0 at both ends, or i = 0, the longest possible wave has the length lo = 2 li, and the frequency /o = -l =--1-, crolo 2 crol1 or n.ny multiple (odd or even) thereof. If then and w again represent the time and the distance angles of the fumlmnentul wave, its harmonics have the respective time and distance angles 2 , 2 w, 3 , 3 w, 4 , 4 w, etc. A complex oscillation then has the form + Cl1 COS ( =t= W - 1'1) a'l COS 2 ( =F w - 'Y2) + a3 cos 3 (ip =F w - 'Ya) + . . . , (38) and the length l1 of the line is represented by angle w1 = 1r, and the oscillation is called a half-wave oscillation. The half-wave oscillation thus contains even as well as odd harmonics, and thereby may have a wave shape, in which one half wave differs from the other. Equations (37) and (38) are of the form of equation (17), but Fig. 48B. - co10049. - Oscillogram of High-frequency Oscillation Preceding Low-frequency OsciHation of Compound Circuit Caused ;; by Switching UH miles of 100,000 Volts Transmission Linc and Step-down Transformer off another 154 1\-Iiles of 100,000 Volts i.:.. Linc; High-tension Switching. ~ .t.-.-. <: Cw ~ ~ l;:j ~ .....,. Fig. 49. -cn10036. -Oscillogmm of Oscillation of Compound Circuit Consisting of 154 Miles of 100,000 Volts Line and Step-up e0o Transformer; Connecting and Disconnecting by Low-tension Switches. High-tension Current and Low-tension Voltage. 84 ELEC7'RJC DISCHARGES, WAVES AND IMPULSES. Substituting u0 = 1 in equations (30) an 1 , and thus prepon 1• Hence it may be expressed in the form: (20) where the value of the power-transfer constant s determines the " st,et'})llPS:'l of wave front." Figs. Gl to rs:3 :,;;how oscillograms of the propagation of such an impul:-;e ov<'r au (artifitial) 1.r:m:-;111i8sio11 li1w of 130 miles,* of the constants: r = 93.6 ohms, L = 0.3944 henrys, C = 1.135 microfarads, \/5 thus of surge impedance zo = = 500 ohms. The impulse is produced by a transformer charge.t Its duration, as measuretl from the oscillograms, is T0 = 0.0030 second. In Fig. 51, the end of the transmission line was connected to a nonin+ d1 sin) sin w + (ca cos 3 + da sin 3 ) e = c 111 sin 3 w + . . . I, I(ai'cos<1>+b1'sin<1>)cosw+(aa'cos3+ba'sin3<1>)1 (4-) 1 cos 3 w + . . . 5. • (c) ARsuming now as instance that, in such a stationary oscilla- tion as giv('ll by equation (47), the current in the circuit is zern at the Rtartiug moment of the transient for = 0. Then the Pquatiou of the current can contain no terms with cos, as these would not be zero for = 0. That is, it must be Ct= 0, Cs= 0, (48) Cr,= 0, At the moment, however, when the current is zero, the voltag~ of the stationary oscillation must be a maximum. As i = 0 for = 0, at this moment the voltage e must be a maximum, that is, t,lie volt.age wave cnn contain no terms with sin, sin 3 , etc. This means b1' = 0, ) ba' = 0, l (49) bs' = 0, etc.) l Substituting (48) and (49) into equation (47) gives i = c•t1 {d1 sin sin w + da sin 3 sin 3 w + ds sin 5 cos w+aa' cos 3 cos 3 w+as' cos 5 =i= w - ,,), f e = eoE-ut sin ( =i= w - i'), (1) where =F w - -y) sin( =F w - -y), = ;e iOE- 2"1 Sin 2 (